标题： 有限范围随机游走在自由群上的通过概率的渐近展开 显示英文标题

标题： Asymptotic Expansion of passage probability for finite-range random walk on Free Groups

摘要： 给定一个在有限生成自由群上的有限范围随机游走，当步数趋于无穷时，该随机游走位于群中某个元素的概率序列的渐近行为是什么？ 在本文中，我们通过提供无限树上不可约有限范围随机游走位于给定顶点的概率序列的任意阶渐近展开式来回答这个问题（见定理6.10）。 我们将假设树的每个顶点的度数大于或等于三，并且假设树被一个保持步长分布的自同构群以余有限的方式作用。 上述问题的答案将作为前述定理的一个直接推论出现。 本文是与[Che25a]和[Che25c]组成的三部曲的一部分。 它依赖于[Che25a]中的Tauberian结果，这些结果在整个文本中用于推导渐近展开式。 本文中引入的对象的详细研究在[Che25c]中进行。 显示英文摘要

摘要： Given a finite-range random walk on a finitely generated free group , what is the asymptotic behaviour, as the number of steps goes to infinity, of the sequence of probabilities that the random walk is at a given element of the group? In this article, we answer this question by providing an asymptotic expansion to any order for the sequence of probabilities that an irreducible finite-range random walk on an infinite tree with bounded valence is at a given vertex (See the Theorem 6.10). We will work under the assumption that each vertex of the tree has valence greater than or equal to three, and we will assume that the tree is endowed with a cofinite action of an automorphism group preserving the step distribution. The answer to the above question will appear as an immediate Corollary of the previously mentioned Theorem. This article is part of a triptych with [Che25a] and [Che25c]. It relies on Tauberian results from [Che25a], used throughout the text to derive asymptotic expansions. The careful study of the objects introduced in this text is done in [Che25c]